An a priori estimate for positive solutions of the Lane-Emden equation in a Lipschitz domain (The structure of function spaces and its environment)

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(1)227. 数理解析研究所講究録第2041巻 2017年 227-233. priori estimate for positive solutions of the Lane‐Emden equation in a Lipschitz domain. An. a. Kentaro Hirata. Department of Mathematics, Graduate School of Science,. Hiroshima \mathrm{E} ‐mail:. University hiratake@hiroshima‐u.ac.jp. Introduction. 1. This note presents Lane‐Emden. improvement of equation, given in many an. related to this. Let $\Omega$ be distance from. a. point. a. bounded. the. x to. positive solutions of the studies. Let start with a simple introduction domain in \mathbb{R}^{n}(n\geq 3) and let $\delta$_{ $\Omega$}(x) denote the usual an a. estimate for. priori. us. boundary \partial $\Omega$ of $\Omega$. .. The Lane‐Emden. equation is a nonlinear. equation of the form. - $\Delta$ u=|u|^{p-1}u. (1.1). ,. where $\Delta$ is the Laplacian on \mathbb{R}^{n} and p>1 We consider the set .. \mathscr{U}_{p}( $\Omega$) ofall positive classical. solutions of (1.1) in $\Omega$ Let .. Ps:=\displaystyle \frac{n+2}{n-2}. It is known that if. and. n. 1<p<p_{S} then there ,. exists. a. positive. constant C. depending only on. u(x)\leq C$\delta$_{ $\Omega$}(x)^{-\frac{2}{p-1} holds for all x\in $\Omega$ and. u\in \mathscr{U}_{p}( $\Omega$). Dancer [2] for the Dirichlet. for. a. property of (1.1): if natural. our. main result,. a\displaystyle \vee b:=\max\{a, b\}. for $\Delta$ For. a. we. growth. we. G(x,y). (1.2) is. some. (1.1). See. -$\Delta$_{q}u=|u|^{p-1}u with $\Delta$_{q}. comes. from the scale invariant. $\lambda$^{\frac{2}{p-1} u( $\lambda$ x)\displaystyle \in \mathscr{U}_{p}(\frac{1}{ $\lambda$} $\Omega$). rate in. need to prepare. \displaystyle\frac{2}{p-1}. on. Poláčik‐Quittner‐Souplet [6]. and Serrin‐Zou [7] for. ,. the. .. ,. ,. and $\lambda$>0 then. for a, b\in \mathbb{R} Let. fixed x_{0}\in $\Omega$. utilized in many studies. .. or not. (1.2). exterior domains,. on. \mathscr{U}_{p}(\mathbb{R}^{n})=\emptyset. was. \mathbb{R}^{n} Note that the exponent‐. u\in \mathscr{U}_{p}( $\Omega$). question whether. To state. .. on. This estimate. .. problem. Liouville type theorem:. being the q ‐Laplacian. and. p. such that. .. Then there arises. optimal?. notations. We write. a\displaystyle \wedge b:=\min\{a,b\}. denote the (Dinchlet) Green function. put. g(x):=G(x,x_{0})\wedge 1.. a. on. $\Omega$.

(2) 228. boundary decay rate of g may vary at each boundary point when \partial $\Omega$ is non‐ smooth, whereas g(x) is comparable to the distance function $\delta$_{ $\Omega$}(x) when \partial $\Omega$ is smooth. We prove the following theorem.. Note that the. Theorem 1.1. Let $\Omega$ be there exists can. a. a. bounded Lipschitz domain in \mathbb{R}^{n}(n\geq 3) and let 1<p<p_{S}. constant C. positive. depending only on. and $\Omega$ such that every. n. p,. .. Then. u\in \mathscr{U}_{p}( $\Omega$). be estimated by. u(x)\displaystyle\leq\frac{C}{g(x)$\delta$_{$\Omega$}(x)^{n-2}\mathrm{v}$\delta$_{$\Omega$}(x)^{\frac{2}{p-1}. for all x\in $\Omega$.. give some remarks For simplicity, we write Here. we. on. the above estimate in. a. (1.3). bounded. Lipschitz. domain $\Omega$.. p_{$\alpha$}:=\displaystyle\frac{n+$\alpha$}{n+$\alpha$-2}. for $\alpha$\geq 0. .. The inverse of g(x)$\delta$_{ $\Omega$}(x)^{n-2} is related to the. functions. on. $\Omega$. .. Indeed,. on a. Martin (Poisson) kernel at .. $\xi$\in\partial $\Omega$. Let. .. $\xi$. .. boundary growth of positive harmonic nontangential region at $\xi$\in\partial $\Omega$ it is comparable to the ,. See Aikawa [1] and the author [3].. It is known that there. are. constants. $\alpha$_{ $\xi$}>0 and. C>1 such that. g(x)\displaystyle\geq\frac{1}{C}$\delta$_{$\Omega$}(x)^{$\alpha$_{$\xi$} on a. if. nontangential region at $\xi$ Therefore, .. (1.4). p<p_{$\alpha$_{ $\xi$}. ,. then. g(x)$\delta$_{ $\Omega$}(x)^{n-2}>$\delta$_{ $\Omega$}(x)^{\frac{2}{p-1} \bullet. on. that set. As. we see. p_{$\alpha$_{ $\xi$}. near. $\xi$ which implies that (1.3) improves the earlier one. ,. from Theorem 3.1 below, the. of(1.1) in $\Omega$ behaving like whether The. growth rate. in (1.3) is. optimal when 1<p< a positive solution. (and also when p_{0}<p<p_{S} because it is known that there is. or. not. \Vert x- $\xi$\Vert^{-\frac{2}{p-1}. near. For. p_{$\alpha$_{ $\xi$} \leq p\leq p_{0}. ,. we. do not know. (1.3) is optimal.. plan of this note is as follows.. In Section 2,. estimates of the Green function and the Martin. of the Newton. $\xi$\in\partial $\Omega$ ).. we. prove Theorem 1.1. kernel,. a. using the global. fundamental pointwise estimate. potential of a superharmonic density and some known results in potential In Section 3, we prove the existence of a positive solution of (1.1) in $\Omega$ behaving theory. like the Martin kernel in order to show that the growth rate in (1.3) is optimal. In the final section, we enumerate some properties one can get from Theorem 1.1..

(3) 229. ProofofTheorem 1.1. 2. In what. follows,. symbol. C,. we. suppose that $\Omega$ is. denote. a. bounded. Lipschitz domain. in. \mathbb{R}^{n}(n\geq 3) By the .. constant whose value may vary at each occur‐. absolute. positive depends on some /\mathrm{a}\mathrm{l}1 of the Lipschitz characters, diam $\Omega$ to specify them. say that C depends on $\Omega$ If necessary, we use C_{1} C2, Let $\xi$\in\partial $\Omega$ and let $\beta$>0. A nontangential region at $\xi$ is defined by we. an. When C. rence.. ,. .. and. $\delta$_{ $\Omega$}(x_{0}). ,. we. \cdots. $\Gamma$_{ $\beta$}( $\xi$):=\{x\in $\Omega$:\Vert x- $\xi$\Vert\leq $\beta \delta$_{ $\Omega$}(x)\}. This set is nonempty and. $\beta$\geq$\beta$_{ $\Omega$} kernel. .. Let. us. recall the. M_{ $\Omega$}(x, $\xi$). at. $\xi$. is accessible from there whenever. global. $\beta$. estimates for the Green function. $\xi$\in\partial $\Omega$ established in [4].. For x,. y\in\overline{ $\Omega$} and C_{1}>1. is. sufficiently large,. ,. we. say. and the Martin. G_{ $\Omega$}(x,y) let. \displaystyle \mathscr{R}(x,y) :=\{b\in\overline{ $\Omega$}:\frac{1}{C_{1} (\Vert x-b\Vert\ve \Vert b-y\Vert)\leq\Vert x-y\Vert\leq C_{1}$\delta$_{ $\Omega$}(b)\}. It is not difficult to. large.. see. that \mathscr{R}(x,y) is nonempty for any pair x,y whenever C_{1} is. Then there exists. a. constant C>1. depending only on. n. sufficiently. and $\Omega$ such that. \displaystyle \frac{1}{C}\frac{g(x)g(y)}{g(b_{xy})^{2} \Vert x-y\Vert^{2-n}\leq G_{ $\Omega$}(x,y)\leq C\frac{g(x)g(y)}{g(b_{xy})^{2} \Vert x-y\Vert^{2-n} for all. (2.1). x,y\in $\Omega$ and b_{xy}\in \mathscr{R}(x,y) ;. \displaystyle \frac{1}{C}\frac{g(x)}{g(b_{x $\xi$})^{2} \Vert x- $\xi$\Vert^{2-n}\leq M_{ $\Omega$}(x, $\xi$)\leq C\frac{g(x)}{g(b_{x $\xi$})^{2} \Vert x- $\xi$\Vert^{2-n} for all x\in $\Omega$ and. depending only. b_{x $\xi$}\in \mathscr{R}(x, $\xi$) on. $\beta$,. n. .. In. particular,. for each. $\beta$\geq$\beta$_{ $\Omega$}. there exists. (2.2). a. constant C>1. and $\Omega$ such that. \displaystyle \frac{1}{C}$\delta$_{ $\Omega$}(x)^{2-n}\leq g(x)M_{ $\Omega$}(x, $\xi$)\leq C$\delta$_{ $\Omega$}(x)^{2-n} for all. x\in$\Gamma$_{ $\beta$}( $\xi$). (see [3]). Also,. we can see. the. following fact from. (2.3) the Harnack. inequality. and the Carleson estimate for positive harmonic functions: .. There exists any pair. Note that. a. constant C>0. x,y\in\overline{ $\Omega$}. and. depending only on b\in \mathscr{R}(x,y). n. and $\Omega$ such that. g(x)\leq Cg(b). for. .. this, together with (2.2) and (2.3), yields that. G_{ $\Omega$}(x,y)\displaystyle \leq C\frac{g(y)}{g(x)}\Vert x-y\Vert^{2-n}. (2.4). M_{ $\Omega$}(x, $\xi$)\displaystyle \leq\frac{C}{g(x)$\delta$_{ $\Omega$}(x)^{n-2}. (2.5). and. for all. x,y\in $\Omega$ Also, .. we use. the. following elementary estimate..

(4) 230. Lemma 2.1. Let. u. be. a. nonnegative superharmonicfunction. \displaystyle \int_{B(x,r)}\frac{u(y)}{\Vert x-y\Vert^{n-2} dy\leq\frac{$\sigma$_{n} {2}r^{2}u(x) where $\sigma$_{n} is the surface area. on. B(x, r) Then .. ,. of the unit sphere in \mathbb{R}^{n}.. Proof. By the polar coordinate representation and the spherical mean value inequality for superharmonic functions, we have. \displaystyle \int_{B(x,r)}\frac{u(y)}{\Vert x-y\Vert^{n-2} dy=\int_{0}^{r}\frac{1}{$\rho$^{n-2} \int_{\partial B(x, $\rho$)}u(y)d $\sigma$(y)d $\rho$ \displaystyle \leq$\sigma$_{n}u(x)\int_{0}^{r} $\rho$ d $\rho$=\frac{$\sigma$_{n} {2}r^{2}u(x). .. \square. We. are now. ready to prove Theorem. Proofof Theorem. 1.1. Let. u\in \mathscr{U}_{p}( $\Omega$). .. 1.1.. As stated in the. (1.2). Therefore it satisfies the differential. introduction, we note that. u. 0\leq- $\Delta$ u(x)\leq C_{2}$\delta$_{ $\Omega$}(x)^{-2}u(x) for all x\in $\Omega$ where C_{2} is ,. have. only. a. positive. such that. (2.6). depending only on p and n To get (1.3), we positive constant C depending only on p, n and $\Omega$. constant. to show that there exists a. .. u(x)\displaystyle \leq\frac{C}{g(x)$\delta$_{ $\Omega$}(x)^{n-2}. holds for all x\in $\Omega$. functions, there. (2.7). decomposition theorem for nonnegative superharmonic exists a nonnegative harmonic function h on $\Omega$ such that .. By the. Riesz. u(x)=h(x)+\displaystyle \int_{ $\Omega$}G_{ $\Omega$}(x,y)(- $\Delta$ u(y) dy for all x\in $\Omega$ Moreover, .. by substituting. x=x_{0} in. (2.8),. we. Let x\in $\Omega$ and let. j\in \mathbb{N}. ,. (2.8). have. \displaystyle \int_{ $\Omega$}g(y)(- $\Delta$ u(y) dy\leq u(x_{0}). (2.9). .. which will be chosen later. We write. B_{j} :=B(x,$\delta$_{ $\Omega$}(x)/2^{j}). plicity. By (2.4), we have. for all. satisfies. inequality. G_{ $\Omega$}(x,y)\displaystyle \leq\frac{2^{j(n-2)}C}{g(x)$\delta$_{ $\Omega$}(x)^{n-2} g(y). y\in $\Omega$\backslash B_{j} Therefore, by (2.9), .. \displaystyle\int_{$\Omega$\backslashB_{j} G_{$\Omega$}(x,y)(-$\Delta$u(y) dy\leq\frac{2^{j(n-2)}C}{g(x)$\delta$_{$\Omega$}(x)^{n-2} u(x_{0}). .. for sim‐.

(5) 231. Since. G_{ $\Omega$}(x,y)\leq C\Vert x-y\Vert^{2-n} and u is superharmonic on $\Omega$. ,. it follows from (2.6) and Lemma. 2.1 that. \displaystyle \int_{B_{j} G_{ $\Omega$}(x,y)(- $\Delta$ u(y) dy\leq\frac{C}{$\delta$_{ $\Omega$}(x)^{2} \int_{B_{j} \frac{u(y)}{\Vert x-y\Vert^{n-2} dy\leq\frac{C_{3} {2^{2j} u(x) where C3. depends only on. h and (2.5),. we. n. p,. and $\Omega$ Moreover, .. by the Martin integral representation of. get. h(x)\displaystyle \leq\frac{C}{g(x)$\delta$_{ $\Omega$}(x)^{n-2} h(x_{0})\leq\frac{C}{g(x)$\delta$_{ $\Omega$}(x)^{n-2} u(x_{0}) These estimates and (2.8). u(x_{0})\leq C by (1.2),. .. yield that. u(x)\displaystyle \leq\frac{2^{j(n-2)}C}{g(x)$\delta$_{ $\Omega$}(x)^{n-2} u(x_{0})+\frac{C_{3} {2^{2j} u(x) Since. ,. we can. obtain (2.7). by choosing j. .. such that. C_{3}/2^{2j}\leq 1/2.. \square. Optimalityofour estimate. 3 The. following theorem shows that the growth rate in (1.2). Theorem 3.1. Let number $\lambda$_{1} such. 1<p<p_{$\alpha$_{ $\xi$}}. ,. is. optimal.. where a_{ $\xi$} is the constant in (1.4). Then there exists there exists a positive classical solution u. thatfor a ny $\lambda$\in(0,$\lambda$_{1} ],. a. positive. of(1.1). in. $\Omega$ such that. \displaystyle \frac{ $\lambda$}{2}M_{ $\Omega$}(x, $\xi$)\leq u(x)\leq\frac{3 $\lambda$}{2}M_{ $\Omega$}(x, $\xi$). (3.1). for all x\in $\Omega$. To show. this,. we. apply the Banach fixed point theorem to. following function class. the. and operator. Let $\lambda$>0 We consider the closed set .. W_{ $\lambda$}:= in the Banach space. { w\displaystyle \in C( $\Omega$):\frac{ $\lambda$}{2}\leq w(x)\leq\frac{3 $\lambda$}{2}. (BC( $\Omega$), \Vert\cdot\Vert_{\infty}) the ,. equipped with the uniform norm,. set of. for x\in $\Omega$. }. all bounded continuous functions. and the operator \mathscr{T}_{ $\lambda$}. on. W_{ $\lambda$} defined by. J_{ $\lambda$}^{ $\sigma$-}[w](x):= $\lambda$+\displaystyle \frac{1}{M_{ $\Omega$}(x, $\xi$)}\int_{ $\Omega$}G_{ $\Omega$}(x,y)(w(y)M_{ $\Omega$}(y, $\xi$) ^{p}dy for x\in $\Omega$. .. Using (2.1) and (2.2), we can show that if 1<p<p_{$\alpha$_{ $\zeta$}}. ,. then. A:=\displaystyle \sup_{x\in $\Omega$}\frac{1}{M_{ $\Omega$}(x, $\xi$)}\int_{ $\Omega$}G_{ $\Omega$}(x,y)M_{ $\Omega$}(y, $\xi$)^{p}dy is finite. See [5] for details.. on. $\Omega$.

(6) 232. T_{ $\lambda$}(W_{ $\lambda$})\subset W_{ $\lambda$} whenever $\lambda$. Lemma 3.2.. Proof.. Let. w\in W_{ $\lambda$} Since p>1 .. ,. we. sufficiently small.. is. get. \displaystyle\frac{$\lambda$}{2}\leq$\lambda$-A(\frac{3$\lambda$}{2})^{p}\leqJ_{$\lambda$}^{$\sigma$-}[w](x)\leq$\lambda$+A(\frac{3$\lambda$}{2})^{p}\leq\frac{3$\lambda$}{2} sufficiently small. Since (w(y)M_{ $\Omega$}(y, $\xi$))^{p} is locally bounded on $\Omega$ the classical result shows that the Green potential of that density is continuous on $\Omega$, \square and so is J_{ $\lambda$}[w] Hence J_{ $\lambda$}^{ $\sigma$-}[w]\in W_{ $\lambda$}. for all x\in $\Omega$ whenever $\lambda$ is ,. ,. .. T_{ $\lambda$}:W_{ $\lambda$}\rightar ow W_{ $\lambda$}. Lemma3.3.. Proof.. Let w_{1},. w_{2}\in W_{ $\lambda$}. .. is a contraction. For x\in $\Omega$. ,. we. mapping whenever $\lambda$. is. sufficiently small.. get. |J_{ $\lambda$}^{-} $\sigma$[w_{1}](x)-T_{ $\lambda$}[w_{2}](x)|\displaystyle \leq\int_{ $\Omega$}\frac{G_{ $\Omega$}(x,y)M_{ $\Omega$}(y, $\xi$)^{p} {M_{ $\Omega$}(x, $\xi$)}|w_{1}(y)^{p}-w_{2}(y)^{p}|dy \leq A\Vert w_{1}^{p}-w_{2}^{p}\Vert_{\infty}.. Since. \displaystyle \Vert w_{1}^{p}-w_{2}^{p}\Vert_{\infty}\leq p(\frac{3 $\lambda$}{2})^{p-1}\Vert w_{1}-w_{2}\Vert_{\infty} by the mean value theorem, we can obtain. |\displaystyle \mathscr{T}_{ $\lambda$}[w_{1}](x)-\mathscr{T}_{ $\lambda$}[w_{2}](x)|\leq\frac{1}{2}\Vert w_{1}-w_{2}\Vert_{\infty} for all x\in $\Omega$ whenever $\lambda$ is small ,. Proofof Theorem 3.1. By the such that \mathscr{T}_{ $\lambda$}[w_{0}]=w_{0}. on. $\Omega$. .. enough.. Thus the lemma follows.. \square. point theorem, there exists u(x):=w_{0}(x)M_{ $\Omega$}(x, $\xi$) we have Letting Banach fixed. a. unique w_{0}\in W_{ $\lambda$}. ,. u(x)= $\lambda$ M_{ $\Omega$}(x, $\xi$)+\displaystyle \int_{ $\Omega$}G_{ $\Omega$}(x,y)u(y)^{p}dy for all x\in $\Omega$. .. Also, (3.1) holds. Since. theorem shows that. u\in C^{2}( $\Omega$). .. Hence. locally bounded on $\Omega$ the classical regularity \square u is a positive classical solution of(1.1) in $\Omega$. u. is. ,. Remark. 4. important and wide applications. Indeed, in potential theory, one can get Theorem 1.1 has. \bullet. a. with the. help. strong Harnack inequality: for each small 0< $\kappa$\ll 1 there exists ,. of some results. a. constant. c(K). depending only on K, p and n such that u(x)\leq c( $\kappa$)u(y) for all u\in \mathscr{U}_{p}( $\Omega$) and any pair x,y\in $\Omega$ satisfying \Vert x-y\Vert\leq $\kappa \delta$_{ $\Omega$}(x) Moreover, c( $\kappa$) enjoys c( $\kappa$)\geq 1 and c(K)\rightarrow 1 .. as. $\kappa$\rightarrow 0+ ;.

(7) 233. \bullet. \bullet. the existence of nontangential limits of a. and the ratio. Harnack convergence theorem: any sequence in. converges \bullet. u\in \mathscr{U}_{p}( $\Omega$). uniformly. to a. function in. \mathscr{U}_{p}( $\Omega$)\cup\{0\}. \mathscr{U}_{p}( $\Omega$). on. has. a. ulM_{ $\Omega$}(\cdot, $\xi$) ;. subsequence which. each compact subset of $\Omega$.. the existence and nonexistence of a positive classical solution of. \left{bgin{ary}l -$\Delta$u=^{p}\ u=$\lambd\elta$_{\xi$} end{ary}\ight.mahr{i}\mathr{n}$\Omega$\mthr{o}\mathr{n}\partil$\Omega$ where $\lambda$>0 and find. a. $\delta$_{ $\xi$}. is the Dirac. measure. (4.1) ,. concentrated at. critical number $\lambda$^{*} such that if $\lambda$\leq$\lambda$^{*}, then (4.1) has. $\lambda$>$\lambda$^{*} then (4.1) has ,. no. These results and their proofs. $\xi$\in\partial $\Omega$ Indeed, .. a. we can. positive solution, but if. positive solution. can. be found in [5].. References [1] H. Aikawa, On the minimal thinness in. a. Lipschitz domain, Analysis. 5. (1985),. no.. 4,. 347‐382.. [2] E. N. Dancer, Superlinear problems. on. exterior problems, Math. Z. 229 (1998),. domains with holes. no.. of asymptotic shape. and. 3, 475−491.. [3] K. Hirata, Estimates for the products of the Green function and the Martin kernel,. Nagoya Math. J.. 188 (2007), 1‐18.. [4] K. Hirata, Global estimates for non‐symmetric Green the p ‐Laplace equation, Potential Anal. 29 (2008),. typefunctions with applications to. no.. 3, 221‐239.. [5] K. Hirata, Existence and nonexistence ofa positive solution of the Lane‐Emden equation. having a boundary singularity:. the subcritical case,. preprint.. [6] P. Poláčik, P. Quittner and P. Souplet, Singularity and decay estimates in superlinear. problems. via. 139 (2007),. Liouville‐type. no.. Elliptic equations. and systems, Duke Math. J.. 3, 555‐579.. Cauchy‐Liouville and universal boundedness theorems for quasi‐ equations and inequalities, Acta Math. 189 (2002), no. 1, 79‐142.. [7] J. Serrin and H. Zou, linear elliptic. theorems. I..

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